Mixing Juice Concentrations

A juice company mixes 10 liters of juice with 30% fruit content with 6 liters of juice with 50% fruit content. What is the fruit concentration in the mixture?

Identify what we know: Solution 1: 10L at 30% = $10 \times 0.30$ Solution 2: 6L at 50% = $6 \times 0.50$ = Pure fruit amounts

Calculate pure fruit in each: $10 \times 0.30 = 3$ L of pure fruit $6 \times 0.50 = 3$ L of pure fruit = 3L + 3L = 6L total pure fruit

Find total mixture volume: $10 + 6 = 16$ liters total = 16 liters

Calculate mixture concentration: $\frac{6}{16} = 0.375 = 37.5\%$ = 37.5% fruit content

Answer: The mixture has a 37.5% fruit concentration

Finding Unknown Volume

A chemist has a 25% acid solution. How many liters of a 60% acid solution must be added to 8 liters of the 25% solution to create a 40% acid solution?

Set up the equation: Let $x$ = liters of 60% solution needed Acid from 25%: $0.25 \times 8 = 2$ L Acid from 60%: $0.60 \times x$ L Total volume: $8 + x$ L = Variables defined

Write the mixture equation: $0.25(8) + 0.60(x) = 0.40(8 + x)$ = Acid in = Acid out

Solve for x: $2 + 0.6x = 3.2 + 0.4x$ $0.6x - 0.4x = 3.2 - 2$ $0.2x = 1.2$ $x = 6$ = $x = 6$ liters

Verify the answer: Check: $0.25(8) + 0.60(6) = 2 + 3.6 = 5.6$ L acid Total: $14$ L at $40\% = 5.6$ L acid = Verified!

Answer: 6 liters of the 60% solution must be added

Coffee Blend Pricing

A coffee shop wants to create 20 kg of a blend that sells for 12 euros per kg. They mix premium coffee at 15 euros/kg with standard coffee at 9 euros/kg. How much of each type should they use?

Define variables: Let $x$ = kg of premium (15 euros/kg) Then $20 - x$ = kg of standard (9 euros/kg) = Two unknowns linked

Set up value equation: Total value: $20 \times 12 = 240$ euros $15x + 9(20-x) = 240$ = Price equation

Solve for x: $15x + 180 - 9x = 240$ $6x + 180 = 240$ $6x = 60$ $x = 10$ = $x = 10$ kg premium

Find standard amount: $20 - 10 = 10$ kg standard = 10 kg standard

Verify: $10 \times 15 + 10 \times 9 = 150 + 90 = 240$ euros $240 \div 20 = 12$ euros/kg = Verified!

Answer: Use 10 kg of premium coffee and 10 kg of standard coffee

Diluting a Solution

A lab has 12 liters of 70% alcohol solution. How much pure water must be added to dilute it to 40% alcohol?

Identify the key insight: Water has 0% alcohol Adding water increases volume but not alcohol amount = Alcohol amount stays constant

Calculate alcohol amount: Alcohol in original: $12 \times 0.70 = 8.4$ L = 8.4 L of pure alcohol

Set up dilution equation: Let $w$ = liters of water New volume: $12 + w$ New concentration: $\frac{8.4}{12 + w} = 0.40$ = Equation ready

Solve for w: $8.4 = 0.40(12 + w)$ $8.4 = 4.8 + 0.4w$ $3.6 = 0.4w$ $w = 9$ = $w = 9$ liters

Verify: New volume: $12 + 9 = 21$ L Concentration: $\frac{8.4}{21} = 0.40 = 40\%$ = Verified!

Answer: Add 9 liters of water to create a 40% alcohol solution

Mistake: Adding percentages directly: 30% + 50% = 80%

Why: Percentages don't add when mixing different volumes. You must weight them by volume.

Correct: Calculate the actual amounts, add those, then find the percentage of the total.

Mistake: Forgetting that water has 0% concentration

Why: When diluting, students often forget to account for water as a solution with 0% solute.

Correct: Water contributes to volume but not to solute amount: $0.00 \times V_{water} = 0$

Mistake: Using the wrong total volume

Why: The mixture volume is the sum of all parts, not just one solution.

Correct: $V_{total} = V_1 + V_2$ (or $V_1 + V_2 + V_3$ for three solutions)

Mistake: Converting percentages incorrectly

Why: 30% must be written as 0.30 (not 30) in calculations.

Correct: Always convert: $30\% = \frac{30}{100} = 0.30$

Pharmacy Compounding

Pharmacists mix medications to create specific concentrations for patients.

A pharmacy needs to prepare 200 mL of a 2% saline solution from a 5% stock solution and distilled water.

Metal Alloy Production

Metallurgists create alloys by mixing metals with different compositions.

Bronze is made by mixing copper (90%) and tin (10%). A factory mixes pure copper with a 70% copper alloy.

Coffee Roasting Business

Coffee roasters blend beans of different prices to create signature blends at target price points.

A roaster creates a house blend by mixing Ethiopian beans (18 euros/kg) with Brazilian beans (10 euros/kg) to sell at 13 euros/kg.

Q: Can mixture problems involve more than two substances?

A: Yes! The same principle applies: $C_1V_1 + C_2V_2 + C_3V_3 = C_{mix}V_{mix}$. Just add more terms for each substance.

Q: What if I'm mixing different types of things (like price and concentration)?

A: The same weighted average approach works! For prices: $P_1Q_1 + P_2Q_2 = P_{mix}Q_{mix}$ where P is price and Q is quantity.

Q: How do I know which variable to solve for?

A: Read carefully: if the problem asks 'how much of solution A' - that's your variable. The unknown quantity is what you solve for.